No, actually, they’re not. They’re anti-immigrant. Along with the whinging about illegal aliens, there is tons of whinging about the visa program, and constant demands to cut down legal immigration as well because they’re stealing jobs from “Real Murcansâ€.
If they were pro-immigrant, they’d be in favor of liberalizing immigration, not closing the doors completely.
That is a rather broad brush...
I have heard Conservatives say they want a moratorium on any new immigration for a decade or so, to better handle what we've already got, and to allow time to develop a more reasonable and enforceable process... A position I can agree with. The system has to be enforceable.
And I have heard many decry H1B visas because They allow foreign workers to take native jobs for pennies on the dollar - The Conservative solution being to allow H1B visas, but at something akin to current market value of the job.
Both of those positions are about fair play and the rule of law. And neither of them are against immigration.